The Overlap Gap Property in Principal Submatrix Recovery

We study support recovery for a $k \times k$ principal submatrix with elevated mean $\lambda/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $\lambda>0$ is a universal constant, and we assume $k = N \rho$ for some constant $\rho \in (0,1)$. We establish that {there exists a constant $C>0$ such that} the MLE recovers a constant proportion of the hidden submatrix if $\lambda {\geq C} \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}}$, {while such recovery is information theoretically impossible if $\lambda = o( \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}} )$}. The MLE is computationally intractable in general, and in fact, for $\rho>0$ sufficiently small, this problem is conjectured to exhibit a \emph{statistical-computational gap}. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}{\rho} \log \frac{1}{\rho} } \ll \lambda \ll \frac{1}{\rho^{1/2 + \varepsilon}}$, the problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $\lambda > 1/\rho$, a simple spectral method recovers a constant proportion of the hidden submatrix.Comment: 42 pages, 1 figur

Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information- theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs.Comment: For sparse PCA and submatrix localization, we determine the information-theoretic threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse based on a conditional second moment method, closing the factor of root two gap in the first versio

Computational Barriers to Estimation from Low-Degree Polynomials

One fundamental goal of high-dimensional statistics is to detect or recover structure from noisy data. In many cases, the data can be faithfully modeled by a planted structure (such as a low-rank matrix) perturbed by random noise. But even for these simple models, the computational complexity of estimation is sometimes poorly understood. A growing body of work studies low-degree polynomials as a proxy for computational complexity: it has been demonstrated in various settings that low-degree polynomials of the data can match the statistical performance of the best known polynomial-time algorithms for detection. While prior work has studied the power of low-degree polynomials for the task of detecting the presence of hidden structures, it has failed to address the estimation problem in settings where detection is qualitatively easier than estimation. In this work, we extend the method of low-degree polynomials to address problems of estimation and recovery. For a large class of "signal plus noise" problems, we give a user-friendly lower bound for the best possible mean squared error achievable by any degree-D polynomial. To our knowledge, this is the first instance in which the low-degree polynomial method can establish low-degree hardness of recovery problems where the associated detection problem is easy. As applications, we give a tight characterization of the low-degree minimum mean squared error for the planted submatrix and planted dense subgraph problems, resolving (in the low-degree framework) open problems about the computational complexity of recovery in both cases.Comment: 38 page

Simultaneously Structured Models with Application to Sparse and Low-rank Matrices

The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise optimal number of measurements (e.g., $\ell_1$ norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with these norms, then we can do no better, order-wise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the $\ell_1$ and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and low-rank tensor completion.Comment: 38 pages, 9 figure

Efficient reconstruction of band-limited sequences from nonuniformly decimated versions by use of polyphase filter banks

An efficient polyphase structure for the reconstruction of a band-limited sequence from a nonuniformly decimated version is developed. Theoretically, the reconstruction involves the implementation of a bank of multilevel filters, and it is shown that how all these reconstruction filters can be obtained at the cost of one Mth band low-pass filter and a constant matrix multiplier. The resulting structure is therefore more general than previous schemes. In addition, the method offers a direct means of controlling the overall reconstruction distortion T(z) by appropriate design of a low-pass prototype filter P(z). Extension of these results to multiband band-limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized, along with generalizations to the multidimensional case. Design examples are included to demonstrate the theory, and the complexity of the new method is seen to be much lower than earlier ones